Newspace parameters
Level: | \( N \) | \(=\) | \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 980.ba (of order \(14\), degree \(6\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.489083712380\) |
Analytic rank: | \(0\) |
Dimension: | \(6\) |
Coefficient field: | \(\Q(\zeta_{14})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Projective image: | \(D_{7}\) |
Projective field: | Galois closure of 7.1.110730297608000.1 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/980\mathbb{Z}\right)^\times\).
\(n\) | \(101\) | \(197\) | \(491\) |
\(\chi(n)\) | \(\zeta_{14}^{6}\) | \(-1\) | \(-1\) |
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
239.1 |
|
−0.623490 | + | 0.781831i | −1.62349 | − | 0.781831i | −0.222521 | − | 0.974928i | −0.900969 | − | 0.433884i | 1.62349 | − | 0.781831i | 0.222521 | + | 0.974928i | 0.900969 | + | 0.433884i | 1.40097 | + | 1.75676i | 0.900969 | − | 0.433884i | ||||||||||||||||||
379.1 | 0.222521 | + | 0.974928i | −0.777479 | − | 0.974928i | −0.900969 | + | 0.433884i | 0.623490 | + | 0.781831i | 0.777479 | − | 0.974928i | 0.900969 | − | 0.433884i | −0.623490 | − | 0.781831i | −0.123490 | + | 0.541044i | −0.623490 | + | 0.781831i | |||||||||||||||||||
519.1 | 0.900969 | + | 0.433884i | −0.0990311 | − | 0.433884i | 0.623490 | + | 0.781831i | −0.222521 | − | 0.974928i | 0.0990311 | − | 0.433884i | −0.623490 | − | 0.781831i | 0.222521 | + | 0.974928i | 0.722521 | − | 0.347948i | 0.222521 | − | 0.974928i | |||||||||||||||||||
659.1 | 0.900969 | − | 0.433884i | −0.0990311 | + | 0.433884i | 0.623490 | − | 0.781831i | −0.222521 | + | 0.974928i | 0.0990311 | + | 0.433884i | −0.623490 | + | 0.781831i | 0.222521 | − | 0.974928i | 0.722521 | + | 0.347948i | 0.222521 | + | 0.974928i | |||||||||||||||||||
799.1 | 0.222521 | − | 0.974928i | −0.777479 | + | 0.974928i | −0.900969 | − | 0.433884i | 0.623490 | − | 0.781831i | 0.777479 | + | 0.974928i | 0.900969 | + | 0.433884i | −0.623490 | + | 0.781831i | −0.123490 | − | 0.541044i | −0.623490 | − | 0.781831i | |||||||||||||||||||
939.1 | −0.623490 | − | 0.781831i | −1.62349 | + | 0.781831i | −0.222521 | + | 0.974928i | −0.900969 | + | 0.433884i | 1.62349 | + | 0.781831i | 0.222521 | − | 0.974928i | 0.900969 | − | 0.433884i | 1.40097 | − | 1.75676i | 0.900969 | + | 0.433884i | |||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
20.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-5}) \) |
49.e | even | 7 | 1 | inner |
980.ba | odd | 14 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 980.1.ba.b | yes | 6 |
4.b | odd | 2 | 1 | 980.1.ba.a | ✓ | 6 | |
5.b | even | 2 | 1 | 980.1.ba.a | ✓ | 6 | |
20.d | odd | 2 | 1 | CM | 980.1.ba.b | yes | 6 |
49.e | even | 7 | 1 | inner | 980.1.ba.b | yes | 6 |
196.k | odd | 14 | 1 | 980.1.ba.a | ✓ | 6 | |
245.p | even | 14 | 1 | 980.1.ba.a | ✓ | 6 | |
980.ba | odd | 14 | 1 | inner | 980.1.ba.b | yes | 6 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
980.1.ba.a | ✓ | 6 | 4.b | odd | 2 | 1 | |
980.1.ba.a | ✓ | 6 | 5.b | even | 2 | 1 | |
980.1.ba.a | ✓ | 6 | 196.k | odd | 14 | 1 | |
980.1.ba.a | ✓ | 6 | 245.p | even | 14 | 1 | |
980.1.ba.b | yes | 6 | 1.a | even | 1 | 1 | trivial |
980.1.ba.b | yes | 6 | 20.d | odd | 2 | 1 | CM |
980.1.ba.b | yes | 6 | 49.e | even | 7 | 1 | inner |
980.1.ba.b | yes | 6 | 980.ba | odd | 14 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 5T_{3}^{5} + 11T_{3}^{4} + 13T_{3}^{3} + 9T_{3}^{2} + 3T_{3} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(980, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} - T^{5} + T^{4} - T^{3} + T^{2} - T + 1 \)
$3$
\( T^{6} + 5 T^{5} + 11 T^{4} + 13 T^{3} + \cdots + 1 \)
$5$
\( T^{6} + T^{5} + T^{4} + T^{3} + T^{2} + T + 1 \)
$7$
\( T^{6} - T^{5} + T^{4} - T^{3} + T^{2} - T + 1 \)
$11$
\( T^{6} \)
$13$
\( T^{6} \)
$17$
\( T^{6} \)
$19$
\( T^{6} \)
$23$
\( T^{6} - 2 T^{5} + 4 T^{4} - 8 T^{3} + \cdots + 1 \)
$29$
\( T^{6} - 5 T^{5} + 11 T^{4} - 13 T^{3} + \cdots + 1 \)
$31$
\( T^{6} \)
$37$
\( T^{6} \)
$41$
\( T^{6} + 2 T^{5} + 4 T^{4} + 8 T^{3} + \cdots + 1 \)
$43$
\( T^{6} - 2 T^{5} + 4 T^{4} - 8 T^{3} + \cdots + 1 \)
$47$
\( T^{6} - 2 T^{5} + 4 T^{4} - T^{3} + 2 T^{2} + \cdots + 1 \)
$53$
\( T^{6} \)
$59$
\( T^{6} \)
$61$
\( T^{6} + 2 T^{5} + 4 T^{4} + 8 T^{3} + \cdots + 1 \)
$67$
\( (T + 2)^{6} \)
$71$
\( T^{6} \)
$73$
\( T^{6} \)
$79$
\( T^{6} \)
$83$
\( T^{6} - 2 T^{5} + 4 T^{4} - 8 T^{3} + \cdots + 1 \)
$89$
\( T^{6} + 2 T^{5} + 4 T^{4} + 8 T^{3} + \cdots + 1 \)
$97$
\( T^{6} \)
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